1. Simulation Modeling and Analysis
Sampling from Probability Distributions 1 1

2. Outline Inverse Transforms for Random Variate Generation
Direct Transform and Convolution
Acceptance-Rejection Technique 2 2

3. Inverse Transforms for Random Variate Generation
Random variates are required to simulate the vagaries of arrivals, service, processing and the like which take place in the real world.
Once a reliable RNG is available, how does one use it to obtain random variates with selected statistical distributions? 3 17

4. Steps in Inverse Transform Method
1.- Determine the cdf for the desired RV X.
2.- Set F(X) = R
3.- Solve F(X) = R for X in terms of R. I.e. X = F-1(R)
4.- Generate the necessary RN sequence Ri and use it to compute corresponding values Xi for i = 1,2,…, n 4 18

5. Inverse Transform Method for the Exponential Distribution
1.- F(x) = 1 - e -  x , x > 0
2.- F(X) = 1 - e -  x = R
3.- X = - (1/) ln (1 - R)
4.- For i = 1,2,…, n
Xi = - (1/) ln (1 - Ri)
Example: Use RAND in Excel to create 1000 PRN’s and transform them to E. Examine your results in Stat::Fit. 5 19

6. Inverse Transform Method for the Uniform Distribution
1.- F(x) = {(x-a)/(b-a) ; 0 for a < x < b
2.- F(X) = (X -a)/(b-a) = R
3.- X = a + (b-a) R
4.- For i = 1,2,…, n
Xi = a + (b-a) Ri
Example: Use RAND in Excel to create 1000 PRN’s and transform them to U. Examine your results in Stat::Fit. 6 20

7. Inverse Transform Method for the Weibull Distribution
1.- F(x) = 1 - e - x/)  , x > 0
2.- F(X) = 1 - e - X/)  = R
3.- X =  [ ln (1 - R)]1/
4.- For i = 1,2,…, n
Xi =  [ ln (1 - Ri )]1/
Example: Use RAND in Excel to create 1000 PRN’s and transform them to W. Examine your results in Stat::Fit. 7 21

8. Inverse Transform Method for the Triangular Distribution
1.- F(x) = {0, for x <0; x2/2 for 0<x<1; 1-(2-x)2/2, for 1 <x<2; 1, for x > 2.
2.- R=X2/2 (0<X<1) and R = 1- (2-X)2/2 (1<X<2)
3.- X = 2R1/2 , (0<R<1/2); X = 2 - (2(1-R))1/2 , (1/2<R<1)
Example: Use RAND in Excel to create 1000 PRN’s and transform them to W. Examine your results in Stat::Fit. 8 22

9. Inverse Transform Method for Empirical Distributions
Similar procedure applies
1.- Determine the empirical cdf F(x)
2.- Generate R
3.- Using the F(X) curve determine the corresponding value of X. 9 23

10. Transform Method for Distributions without Closed Form Inverse
Normal, gamma and beta distributions have no closed form inverses. Inverse transform method is applicable only approximately.
For example, for the standard normal distribution
X ~ (R (1-R) 0.135)/0.1975
Example: Use RAND in Excel to create 1000 PRN’s and transform them to N. Examine your results in Stat::Fit. 10 24

11. Inverse Transform Method for Discrete Distributions
Similar method applies.
Lookup table
Algebraic 11 25

12. Direct Transform for the Normal
For the normal distribution
F(x) = -x (1/(2)1/2) e -(t2/2) dt
Box-Muller method (pp )
Z1 = (-2 ln R1)1/2 cos(2  R2)
Z2 = (-2 ln R1)1/2 sin(2  R2)
X1 =  +  Z1
X2 =  +  Z2 12 26

13. Convolution Method
Convolution: the probability distribution of two or more independent RV
Useful for Erlang and binomial variates
Erlang w/parameters (K,
Sum of K independent exponential RV each with mean 1/K
X = -(1/ K ) ln  Ri 13 27

14. Acceptance-Rejection Technique
Say random variates X are needed uniformly distributed between 1/4 and 1
Steps
1.- Generate R
2a.- If R > 1/4 make X = R and go to 3
2b.- If R < 1/4 go to 1 until X is obtained
3.- Repeat from 1 for another X 14 28

15. Acceptance-Rejection Technique for Poisson Distribution
Recall Poisson and exponential are closely related.
P(N=n) = e   n/n!
Steps
1.- Set n =0, P=1
2.- Generate R and replace P by P R
3.- If P < e , accept N = n, otherwise go to 2 15 29

16. Acceptance-Rejection Technique for Gamma Distribution
Steps
1.- Compute a = (2-1)1/2 ; b = 2 - ln 4 + 1/a
2.- Generate R1, R2
3.- Compute X =  [R1/(1-R1)]a
4a.- If X > b-ln(R12R2) reject and go to 2
4b.- If X < b-ln(R12R2) use X as is or as X/ 16 30